DP-colorings of uniform hypergraphs and splittings of Boolean hypercube into faces

Abstract

We develop a connection between DP-colorings of k-uniform hypergraphs of order n and coverings of n-dimensional hypercube by pairs of antipodal (n − k)-dimensional faces. Bernshteyn and Kostochka established that the minimum number of edges in a non-2-DPcolorable k-uniform hypergraph is 2 k−1 . In this paper we use the fact that this bound is attained if and only if there exists a splitting of the n-dimensional Boolean hypercube into 2 k−1 pairs of (n − k)-dimensional faces. We give a construction of such splittings for k = 3 t . For k = 5 we prove that such splitting do not exist. Consequently every 5-uniform hypergraph with 16 (or less) edges is 2-DP-colorable.

Stability of Boolean function classes with respect to clones of linear functions

“…Proof. We follow the proof technique of Potapov [16,Proposition 11]. Note that Ω 00 ⊆ Ω = , so every function in D k ∩ X 1 ∩ Ω 00 is even.…”